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The two sides of the residue pairing have the same dimension

Statement

Assume the Axiom of Choice as inherited from the duality suppliers. Let $C$ be a smooth proper geometrically integral curve over a perfect field $k$ and let $L$ be an invertible sheaf on $C$ . Then $\dim_{k} H^{1} (C, L) = \dim_{k} H^{0} (C, ω_{C} \otimes L^{- 1});$ both spaces are finite-dimensional and the equality is the numerical form of Serre duality for $L$ , obtained from the published duality theorem for smooth projective varieties after Every smooth proper curve admits a projective embedding makes $C$ projective. Combined with A nonzero global dual section detects a cohomology class this shows that the residue pairing of The residue pairing of a line bundle with the dual canonical twist is a perfect pairing over perfect $k$ .

Facts & Assumptions

Given: the Axiom of Choice; a perfect field $k$ ; a smooth proper geometrically integral curve $C$ over $k$ ; and an invertible $O_{C}$ -module $L$ .

[F1]

The Axiom of Choice: every family of nonempty sets has a choice function (The Axiom of Choice).

[F2]

$ω_{C} = Ω_{C / k}^{1}$ is the canonical bundle, an invertible $O_{C}$ -module; and $C$ is a smooth curve over $k$ , so it is a $k$ -scheme of finite type whose underlying space has chain dimension one, with smooth structure morphism, and it is proper over $k$ (Canonical bundle and canonical divisors, Invertible sheaves).

[F3]

An invertible $O_{X}$ -module $L$ is locally free of rank one; its dual $L^{\lor} = H o m_{O_{X}} (L, O_{X})$ is again invertible of rank one, and $L^{\lor} \otimes_{O_{X}} L$ is canonically $O_{X}$ . The sheaf $L^{- 1}$ is the dual $L^{\lor}$ , and the tensor product of $O_{X}$ -modules is the sheafification of the componentwise tensor presheaf (Invertible sheaves, Locally free sheaves of finite rank, Tensor product of sheaves of modules).

[F4]

The residue pairing $⟨ -, - ⟩ ⁣ : H^{1} (C, L) \times H^{0} (C, ω_{C} \otimes L^{- 1}) \to k$ of The residue pairing of a line bundle with the dual canonical twist is $k$ -bilinear, and the induced $k$ -linear map $Φ ⁣ : H^{0} (C, ω_{C} \otimes L^{- 1}) \to H^{1} (C, L)^{*}$ , $s \mapsto (c \mapsto ⟨ c, s ⟩)$ , is injective; equivalently, whenever the two sides are finite-dimensional of equal dimension, $Φ$ is an isomorphism (A nonzero global dual section detects a cohomology class).

[F5]

Let $X$ be a smooth projective $k$ -scheme of pure dimension $n$ and let $E$ be a finite locally free $O_{X}$ -module. With $ω_{X} = ⋀^{n} Ω_{X / k}^{1}$ there is a normalized trace $t_{X} ⁣ : H^{n} (X, ω_{X}) \to k$ , independent of a projective embedding, such that for every $0 \leq q \leq n$ the cup product, contraction and trace give a functorial perfect pairing of finite-dimensional $k$ -vector spaces $H^{q} (X, E) \times H^{n - q} (X, E^{\lor} \otimes ω_{X}) ⟶ H^{n} (X, ω_{X}) \overset{t_{X}}{\to} k;$ outside $0 \leq q \leq n$ the relevant cohomology groups vanish (Serre duality for locally free sheaves on a smooth projective variety).

[F6]

Every smooth proper geometrically integral curve $C$ over a field $k$ admits a closed immersion $i ⁣ : C \to P_{k}^{N}$ over $k$ ; equivalently the structure morphism $C \to Spec k$ is projective in the H-projective convention (Every smooth proper curve admits a projective embedding).

Proof

Proof technique: direct; apply the published smooth-projective duality theorem to the curve and the rank-one module $L$ in degree $q = 1$ , and combine injectivity with equal finite dimensions.

1.1F2F5F6

By [F6] the curve $C$ is projective over $k$ , and it is smooth over $k$ with underlying space of dimension one, so $X = C$ satisfies the hypotheses of the duality theorem [F5] with pure dimension $n = 1$ ; moreover $ω_{C} = ⋀^{1} Ω_{C / k}^{1}$ agrees with the canonical bundle of [F2].

1.2F3

The invertible sheaf $L$ is locally free of rank one, hence a finite locally free $O_{C}$ -module of rank $r = 1$ as required for the module $E = L$ in [F5].

2.1F5step 1.1step 1.2

Apply the duality theorem [F5] to $X = C$ , $n = 1$ , $E = L$ and $q = 1$ : the cup product, contraction and normalized trace $t_{C} ⁣ : H^{1} (C, ω_{C}) \to k$ give a perfect $k$ -bilinear pairing $H^{1} (C, L) \times H^{0} (C, L^{\lor} \otimes ω_{C}) ⟶ H^{1} (C, ω_{C}) \overset{t_{C}}{\to} k,$ and both $k$ -vector spaces are finite-dimensional; perfectness makes the induced map to the dual an isomorphism, so their $k$ -dimensions are equal.

3.1F3step 2.1

The dual $L^{\lor}$ is the inverse $L^{- 1}$ of $L$ in the Picard group of invertible sheaves, so the invertible sheaves $L^{\lor} \otimes ω_{C}$ and $ω_{C} \otimes L^{- 1}$ are canonically isomorphic; consequently their spaces of global sections are $k$ -linearly isomorphic.

4.1F5step 2.1step 3.1

Combining steps 2.1 and 3.1, $\dim_{k} H^{1} (C, L) = \dim_{k} H^{0} (C, ω_{C} \otimes L^{- 1})$ , both spaces finite-dimensional: this is the numerical form of Serre duality for the invertible sheaf $L$ on the curve.

5.1F1F4step 4.1∎

The map $Φ ⁣ : H^{0} (C, ω_{C} \otimes L^{- 1}) \to H^{1} (C, L)^{*}$ induced by the residue pairing is $k$ -linear and injective by [F4], and step 4.1 exhibits its source and target as finite-dimensional $k$ -vector spaces of the same dimension; an injective linear map between such spaces is an isomorphism, hence every nonzero class in $H^{1} (C, L)$ is detected by a global section and the residue pairing is perfect. The only choice-theoretic input is the Axiom of Choice inherited through the duality and residue suppliers [F1].

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Sources

John Tate, Residues of differentials on curves, Ann. Sci. E.N.S. (4) 1 (1968) 149-159 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (version of October 21, 2025) (standard reference, not scraped)
MIT 18.725 Algebraic Geometry (Fall 2015) course notes, Lectures 24-25 (standard reference, not scraped)